Results 1 to 10 of 10

Math Help - Help with Identity proof Please

  1. #1
    Newbie
    Joined
    Mar 2010
    Posts
    21

    Unhappy Help with Identity proof Please

    Hello, I am working on a problem in my homework where I need to prove that the identity equation is true. The parameters are, however, that I must only work on one side of the equation. The Equation:

    sec^2x + csc^2x = sec^2x*csc^2x

    I chose the left side to work in. First, using Pythagorean Identity, expand sec and csc:

    (1+tan^2x) + (1+cot^2x)

    Combine the constants:

    tan^2x + cot^2x + 2

    Use the co-function identities to expand again:
    sin^2x/cos^2x + cos^2x/sin^2x + 2/2

    Here is where I think I am stuck, or maybe I took a wrong turn before this. I could go further, and break the sin into its Pythagorean identity:

    (1-cos^2x)/cos^2x + cos^2x/(1-cos^2x) + 2/2

    But then what? No common denominator. I could take the reciprocal of the second term and then add, but it doesn't get me much.

    (1-cos^2x)/cos^2x + (1-cos^2x)/cos^2x + 2/2

    Brings up
    (2-cos^2x)/cos^2x

    This equals 1, but then I still have that 2/2 to add in, so then I get 3. which is not what sec^2x*csc^2x is.

    Where have I gone wrong? (if I've posted this incorrectly, I'll edit it until I get it right. Please forgive any incorrect formatting. This is my first time here.)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: Help with Identity proof Please

    Quote Originally Posted by DamenFaltor View Post
    Hello, I am working on a problem in my homework where I need to prove that the identity equation is true. The parameters are, however, that I must only work on one side of the equation. The Equation:

    sec^2x + csc^2x = sec^2x*csc^2x

    I chose the left side to work in. First, using Pythagorean Identity, expand sec and csc:

    (1+tan^2x) + (1+cot^2x)

    Combine the constants:

    tan^2x + cot^2x + 2

    Use the co-function identities to expand again:
    sin^2x/cos^2x + cos^2x/sin^2x + 2/2

    Here is where I think I am stuck, or maybe I took a wrong turn before this. I could go further, and break the sin into its Pythagorean identity:

    (1-cos^2x)/cos^2x + cos^2x/(1-cos^2x) + 2/2

    But then what? No common denominator. I could take the reciprocal of the second term and then add, but it doesn't get me much.

    (1-cos^2x)/cos^2x + (1-cos^2x)/cos^2x + 2/2

    Brings up
    (2-cos^2x)/cos^2x

    This equals 1, but then I still have that 2/2 to add in, so then I get 3. which is not what sec^2x*csc^2x is.

    Where have I gone wrong? (if I've posted this incorrectly, I'll edit it until I get it right. Please forgive any incorrect formatting. This is my first time here.)
    I started from LHS.

    \csc^2x\cdot \sec^2x

    \csc^2x \cdot (1 + \tan^2x)

    \csc^2x+\csc^2x \cdot \tan^2x

    {\csc^2x+ \frac{1}{\sin^2x}} \cdot \frac{\sin^2(x)}{\cos^2(x)}

    \csc^2x+\frac{1}{\cos^2x}

    \csc^2x+\sec^2x
    Follow Math Help Forum on Facebook and Google+

  3. #3
    -1
    e^(i*pi)'s Avatar
    Joined
    Feb 2009
    From
    West Midlands, England
    Posts
    3,053
    Thanks
    1

    Re: Help with Identity proof Please

    Your 2 has become 2/2 = 1 between your second and third lines of working.

    Multiply throughout by \dfrac{\sin^2(x)\cos^2(x)}{\sin^2(x)\cos^2(x)}

    = \dfrac{\sin^4(x) + \cos^4(x) + 2\sin^2(x)\cos^2(x)}{\sin^2(x)\cos^2(x)}


    The numerator will factor (it's a perfect square) to a form where you can use an identity
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Mar 2010
    Posts
    21

    Re: Help with Identity proof Please

    Ok I see you are going from the right hand side, the multiplication side. But.... How did you go from
    \csc^2x \cdot (1 + \tan^2x)

    to

    \csc^2x+\csc^2x \cdot \tan^2x

    Which identity is that??
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Mar 2010
    Posts
    21

    Re: Help with Identity proof Please

    Ack nevermind, you just multiplied through... man why didn't I see that??
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Mar 2010
    Posts
    21

    Re: Help with Identity proof Please

    Thank you guys! This helped me a lot.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Mar 2010
    Posts
    21

    Re: Help with Identity proof Please

    Also sprach Zarathustra, This is kind of a odd question - but how did you know to pick the sec function to transform but leave the cosecant function in its existing state? Was it just a leap of intuition??
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    May 2011
    From
    Islamabad
    Posts
    94

    Re: Help with Identity proof Please

    if you chose left hand side

    sec^2 x + cosec^2 x = ( 1/cos^2 x ) + (1/sin^2 x) => [(sin^2 x) + (cos^2 x)] / (cos^2 x)(sin^2 x)

    = 1 / (cos^2 x)(sin^2 x) => (sec^2 x)(cosec^2 x)
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: Help with Identity proof Please

    Quote Originally Posted by DamenFaltor View Post
    Also sprach Zarathustra, This is kind of a odd question - but how did you know to pick the sec function to transform but leave the cosecant function in its existing state? Was it just a leap of intuition??
    I think that one answer to this question could be: experience that comes after solving a bunch of this kind of questions...
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Mar 2010
    Posts
    21

    Re: Help with Identity proof Please

    Thank you everyone for your help. I'm trying to study for a test that I have coming up next week, and I am practicing like crazy. My math anxiety is huge, so seeing how these things can get broken down into manageable pieces helps a lot.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof of Set Identity
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: October 13th 2010, 06:21 AM
  2. Identity proof
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: May 19th 2010, 06:31 AM
  3. Another identity proof.
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: May 5th 2010, 11:49 AM
  4. identity proof
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 24th 2009, 09:23 AM
  5. Identity proof
    Posted in the Algebra Forum
    Replies: 5
    Last Post: January 14th 2006, 11:13 PM

Search Tags


/mathhelpforum @mathhelpforum